AdS CFT
The AdS/CFT correspondence is an implementation of the Holographic Principle relating String Theory in Anti-de Sitter Space to a Conformal Field Theory on the conformal boundary of that space. Intuition Consider the Holographic Principle. The principle tells us that all information stored within a region is encoded on it's boundary. See Holographic Principle for the intuition behind this principle. Now, this applies to any region, including an Anti-de Sitter Space. The Anti-de Sitter Space has a conformal boundary, on which a Conformal Field Theory,or CFT, is defined. Therefore, a Quantum Gravitational Theory in Anti-de Sitter Space, or AdS (space) for short, is equivalent to a Conformal Field Theory on it's Conformal Boundary.. Examples Maldacena's Original Example Maldacena's paper discusses three examples of AdS/CFT duality. First and best-known is the duality between Type IIB String Theory on AdS_5 \times S^5 , and N=4 Super-Yang-Mills Theory. Also considered were M-Theory on AdS_4 \times S^7 , which is dual to ABJM Theory (but the equations for ABJM Theory were not figured out until 2008), and M-Theory on AdS_7 \times S^4 , which is dual to (2,0) Theory. Consider a stack of N D3 Branes. They form Yang-Mills Supermultiplets with 4-Dimensional \mathcal{N} = 4 Supersymmetry. The vector hypermultiplets are transforming under a gauge group U(N)\cong SU(N)\times U(1) . In the infrared, the graviational dynamics and strings decouple, and the U(1) hypermultiplet decouples, while the SU(N) hypermultiplets remain interacting as the beta function is zero. Notice that we know have a Conformal Field Theory with \mathcal{N}=4 Supersymmetry. Note that this metric background is that of an extremal 3-Brane black hole. The distance to the event horizon is logarithmically divergent, so it is infinitely far away. As one approaches the event horizon, the geometry approaches AdS_5 \times S^5 . This is the original motivation for AdS/CFT . AdS5/CFT4 example The Stress-Energy Operator on the Conformal Field Theory is dual to the transverse components of the metric on the Anti-de Sitter. The Conformal Boundary Parameterise the Anti-de Sitter Space as follows: ds^2 = (kz)^{-2}\left( dz^2 + \eta_{\mu\nu} \, dx^\mu \, dx^\nu \right) Apply a Conformal Transformation , we get \mathrm{d}s^2 = \mathrm{d}z^2 + \eta_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu Which becomes the Minkowski Metric when z=0 , which is the boundary. This is called the Conformal Boundary. Mathematical Formulation We deform the Conformal Field Theory by adding the Source Fields \int \mbox{d}x^D j_{CFT}(x)\mathcal{O}(x) This is now dual to a theory on Anti-de Sitter Space with a bulk field j . This space would have the boundary condition \lim\limits_{\mathrm{boundary} } j \omega^{\Delta-d+n} = J_{\text{CFT}} Here, \Delta is the Conformal Dimension of the Gauge-Invariant Operator \mathcal{O} and n is the number of covariant indices of \mathcal{O} minus the number of contravariant indices. \left\langle \mathcal{T}\left\{ \exp\left(\int \mathrm{d}x^D J_{4D}(x)\mathcal{O}(x)\right) \right\} \right\rangle_{\mathrm{CFT}} = Z_{\mathrm{AdS}}\leftJ \omega^{\Delta-D+n } = J_{4D}\right The LHS is the Vacuum Expectation Value of the time-ordered exponential of the operators over the Conformal Field Theory. The right hand side is the generating functional for the Quantum Gravity with the given Conformal Boundary Condition. The RHS is evaluated by finding the classical solutions to the effective action subject to the given boundary conditions. Black Hole Firewalls AdS CFT strongly suggests that the formation and evaporation of a black hole are unitary. It has however been argued , , , that this does not allow for a smooth event horizon to a black hole. The argument for this goes that the CFT lacks certain operators that describe the interior. However, in , such operators were constructed by Suvrat Raju and Kyriakos Papadodimias. Related dualities AdS/CMT AdS/CMT is an application of AdS/CFT to apply String Theory to Condensed Matter Physics. See . AdS/QCD AdS/QCD is a generalisation of AdS/CFT in which the "CFT" is not a real CFT, but instead Quantum Chromodynamics, which is not a Conformal Field Theory, but still a Quantum Field Theory. See and Kerr/CFT Fluid/Gravity Category:String Theory